Anti-Zeno purification of spin baths by quantum probe measurements

The quantum Zeno and anti-Zeno paradigms have thus far addressed the evolution control of a quantum system coupled to an immutable bath via non-selective measurements performed at appropriate intervals. We fundamentally modify these paradigms by introducing, theoretically and experimentally, the concept of controlling the bath state via selective measurements of the system (a qubit). We show that at intervals corresponding to the anti-Zeno regime of the system-bath exchange, a sequence of measurements has strongly correlated outcomes. These correlations can dramatically enhance the bath-state purity and yield a low-entropy steady state of the bath. The purified bath state persists long after the measurements are completed. Such purification enables the exploitation of spin baths as long-lived quantum memories or as quantum-enhanced sensors. The experiment involved a repeatedly probed defect center dephased by a nuclear spin bath in a diamond at low-temperature.


Supplementary Note 1. DYNAMICS OF PROBE ELECTRON SPIN COUPLED TO NUCLEAR SPIN BATH
A. Probe-bath interaction Hamiltonian in the star configuration for non-interacting nuclear spins In the absence of an external magnetic field, upon neglecting the zero-field splitting, the interaction Hamiltonian for the probe spin with Pauli operator ⃗ S resonantly coupled to a collection of non-interacting nuclear spins with angular momentum operators { ⃗ I k } is given in the laboratory frame, by Here S is located at the origin, r k is the k th nuclear position, the dipole-dipole couplings are γ k g k , where g k = γ e /r 3 k , γ e , γ k denote the magnetic moments of the electron and the k th nuclear spin respectively [1].
Since the degeneracy in the NV electron spin-space is not broken by any external field, we can choose the z-axis of our laboratory frame to be parallel to the quantization axis of S. With this choice, the Hamiltonian in Supplementary Eq. (1) becomes whereŜ z , I z k and r z k denote z components of the vectors S, I k and r k respectively. Defining a new vector g k (r) by the components of the hyperfine coupling in Supplementary Eq. (2), we arrive at the Hamiltonian in Eq. 1 of the Main Text. A factor of 1 2 obtained from the representation ofŜ z in the |e(g)⟩ basis is absorbed in g k (r).

B. Conditional evolution of a non-interacting spin bath
The time evolution operator under the interaction Hamiltonian H in Eq. 1 of the Main Text is simply U S+B (t) = exp[−i H t]. For the non-interacting spin bath and in the absence of an external magnetic field, the basis |±⟩ = 1 √ 2 |e⟩±|g⟩ used in the Main Text is degenerate, so that the probe undergoes pure dephasing under the influence of the bath noise, i.e. random mixing of the |+⟩ and |−⟩ states. The time evolution operator, in this case, can be written as The initial probe density matrix is ρ S (0) = |+⟩⟨+|, while the initial bath density matrix ρ B (0) is given by Eq. 4 of the Main Text. The density matrix of the probe + bath, ρ(t), at time t can be written as S projection onto the |+⟩ state yields the conditional bath evolution Similarly, S projection onto the |−⟩ state yields the conditional bath evolution C. Probe-bath interaction Hamiltonian for an interacting-spin bath For a probe spin resonantly coupled to one of the spins (labeled as 1) of a dipole-dipole interacting-spin bath, the full Hamiltonian has the following form, upon taking the energy origin to be at the resonant frequency where S 1 is the Pauli operator pertaining to spin 1, g 1 is the corresponding probe-spin dipolar coupling and the second term describes the interaction of spins i, j, via the dipolar coupling J i,j . In principle, one can always diagonalize the second term and transform the Hamiltonian to the simplified form [2] where J is the effective coupling, c † S is the fermionic probe-excitation (raising) operator, {b k } are the bath-eigenmode deexcitation (lowering) operators and ϕ k are the corresponding expansion coefficients. For the nearest-neighbour spin-spin interaction in a one-dimensional spin-chain configuration known as the magnon model, ϕ k = 2 N +1 sin kπ N +1 (cf. ref. 3 of the Main Text). For more intricate spin-bath geometries there are no general expressions for ϕ k and the corresponding spin-bath eigenvalues ϵ k , but they are obtainable numerically.

D. Conditional evolution of an interacting-spin bath
The degeneracy of the symmetric and antisymmetric superposition states of the probe |±⟩ = 1 √ 2 [ |e⟩ + |g⟩ ] is broken by coupling to an interacting-spin bath, so that the probe no longer undergoes pure dephasing due to couplings to such a bath. We may now measure the probe in the |e(g)⟩ basis, thereby projecting the S+B state onto the corresponding bath state at time t = τ . Thus, for |e⟩ or |g⟩ projections, ρ e(g) We may also project the probe on the |±⟩ basis, even though it is now non-degenerate.
The steering of the bath evolution via a sequence of |+⟩ and |−⟩ projections or |e⟩ and |g⟩ projections defines CTs analogous to those discussed in the Main Text. Yet the resulting bath dynamics and its steady states may be drastically different (see Sec. IV).

Supplementary Note 2. SELECTIVE (SINGLE-SHOT) PROBE MEASUREMENT EFFECTS ON A NON-INTERACTING BATH VIA THE KK FORMULA
If we perform a selective measurement on the probe spin e.g. project it onto the basis state |+⟩, the bath state collapses to (see Main Text) Similarly, for a projection onto the |−⟩ state, the bath state collapses to A conditional trajectory (CT) in parameter space is here defined by a successful sequence of m projections of the probe, n out of which are on the |−⟩ state and m − n on the |+⟩ state. As long as we keep τ fixed, the order of projections on |+⟩ or |−⟩ is immaterial, i.e. they can be permuted at will. For such a CT the final density matrix of the bath is given by, N n,(m−n) (τ ) being the overall normalization factor for this CT, which is denoted by M n,m in Eq. 8 of the Main Text.
The probability distribution for the bath state modified by this CT is The distribution ofP j (τ ) is approximately the bath spectrum P (ω j ) in Fig. 1(c). The cosine-sine rational polynomial in Supplementary Eq. (13) plays the role of the filter function F (τ, ω) in the KK formula (Eq. 10 in the Main Text).

A. Numerical results
We consider an N -spin-1/2 bath and m single-shot measurements. In order to use dimensionless parameters for simulation, we scale the bath-state eigenvalues by the rms couplinḡ g = k g 2 k . The bath eigenfrequencies are to be determined from linear combinations of the couplings g k that are typically random. Supplementary Fig.1 and Supplementary Fig.2 show the bath spectra for N = 4 and N = 10 spins after m = 28 and m = 100 measurements, respectively. The top left panels are for n = 0 (i.e. all |+⟩ detection), for different values ofḡτ . In Supplementary Fig.1 we have assumed that ω j g vary linearly within a period of −π ≤ ω j g ≤ π in steps of π/n b , where n b = 2 N is the bath Hilbert-space dimension. In the top left panel of Supplementary Fig.1 we see that forḡτ ≥ 0.1, the bath-spectrum becomes increasingly pure with the growth ofḡτ . The maximum peak amplitude is obtained forḡτ = 0.8. This signifies an Anti-Zeno (AZE) cooling of the bath spectrum. Supplementary Fig.1 is well in agreement with the experimentally obtained post-measurement ODMR spectrum shown in 2(c) of the Main Text. In Supplementary Fig.1 and Supplementary  Fig.2 ( top right panels) we find that in the AZE limit, as n increases, the two major peaks in the purified bath spectrum shift from the center, toward the spectral edges. For some intermediate values, we have more than two peaks with reduced overall purity.
We repeat this analysis in the Zeno regime withḡτ = 0.05 and show the resulting spectra in Supplementary Fig.1 (bottom panel). Notice that in this case the orange curve almost overlaps the black curve, as confirmed by Supplementary Eq. (16) below. For n ̸ = 0, the spectrum again becomes highly pure with increasing n. However, in this case, the peaks appear near the spectral edge. Similar spectra for an N = 10 spin-1/2 bath is shown in Supplementary Fig.2.
In order to understand the nature of this spectrum for different values ofḡτ we compare the behaviour of cos(x) and x in the range 0 ≤ x ≤ π as shown in Supplementary Fig.3. The graphical solution of cos(x) = x yields x ≈ 0.74 for x ≥ 0. For |x| ≪ 0.74, cos(x) → 1.
In Supplementary Fig.1, forḡτ = 0.8, we have 2  Gaussian envelope enhances the contrast between the peak and other values. This is the Anti-Zeno (AZE) regime of measurement intervalsḡτ . The opposite Zeno (QZE) limit is obtained forḡτ ≪ 0.74. In this case for n = 0 (i.e. all |+⟩ detection), in the QZE limitḡτ → 0, we have from Supplementary Eq. (14) and i.e. the bath spectrum remains unchanged (frozen) throughout the CT as shown in Supplementary Fig.1. This is the Zeno (QZE) regime of measurement intervalsḡτ . Interestingly, in both the AZE and QZE regimes of the measurement intervalsḡτ described above, the bath-spectra depend on the CT through n, the number of |−⟩ detections.
In this case we have negligible spectral contributions for 2 ω j g ḡτ → 0. In Supplementary  Fig.1, forḡτ = 0.05 (QZE regime ofḡτ ), the edge states have the highest non-negligible value: 2 ω j g maxḡ τ ∼ 0.32 < 0.74. Hence these states should have the highest spectral amplitudes. All other frequencies ω j g have even lower values of the argument 2 ω j g ḡτ , leading to blackuced spectral amplitudes. Yet this does not fully explain the dramatic enhancement of the edge-state amplitudes observed in Supplementary Fig.1 and Supplementary Fig.2 in the QZE regime ofḡτ . Moreover, the overall Gaussian envelope indicates further blackuction of the edge-state amplitudes.
The key feature that results in the spectacular enhancement of the edge-state amplitudes in the QZE regime ofḡτ is the normalization factor N n,0 (τ ), resulting from selective singleshot measurements. Since the major contribution to N n,0 (τ ) comes from the edge-states in this case, normalization with N n,0 (τ ) shoots up the edge-state spectral amplitudes resulting in the observed dramatic enhancement in Supplementary Fig.1 and Supplementary Fig.2. Forḡτ = 0.8 and m = n (AZE regime ofḡτ ) in Supplementary Fig.1, 2 ω j g maxḡ τ ≈ 5 ≫ 0.74. Thus in this case the edge-states may or may not have the highest contribution to the spectrum. Consequently, the spectral peaks appear at intermediate positions, jointly determined by the Gaussian envelope P j (0), the normalization factor N n,0 and the filter

Supplementary Note 4. FREE INDUCTION DECAY MODIFICATION IN THE ZENO AND ANTI-ZENO REGIMES
Here we calculate the Free Induction Decay (FID) signal obtained from probe measurements for a probe initialized to |+⟩. Using the time evolution operator U S+B , we can find the joint state of the probe and bath at any time t. Tracing out the bath variables, we obtain the probe density matrix ρ S (t). Projection of ρ S (t) on |+⟩⟨+| then results in the FID that expresses the coherence decay probability. Prior to the CT measurement sequence, this probability is determined by the initial j-state probabilities P j (0) (see Supplementary Eq. (16)) as Since the P j (0) are drawn from a Gaussian distribution, the FID is then the Gaussian average of cos 2 ω j t . Thus, the pre-measurement (prior) FID is given by where T 2 is a constant depending on ω k -s, and other spectral parameters and ... This becomes more evident if we calculate the corresponding curve for a larger value ofḡτ as shown in the right panel of Supplementary Fig.4. The FID obtained in the Zeno-regime is also shown in Supplementary Fig.4. The FID obtained for an N = 10 spin-1/2 bath is shown in Supplementary Fig.5.

Supplementary Note 5. CORRELATION MEASUREMENTS
The probability of a |+⟩-state detection at time t d , long after the bath has reached a steady-state following the CT M n,(m−n) , is where P j (m, n) = P j (0) cos 2(m−n) (ω j τ ) sin 2n (ω j τ ). Similarly the probability of |−⟩ detection is At sufficiently long times, t d ≥ Γ −1 , the second terms in the expressions for p ± become negligible. Thus, in this limit we have We have seen thus far (SI Sec. III-IV) that the Anti-Zeno (AZE) polarization of the bath in the absence of an external field and for a non-interacting spin bath leads to a maximum bath polarization of 0.5, on account of the |± state symmetry of the probe. An example Such states have no practical application in quantum sensing. For sensing applications we may resort to the intra-bath couplings (SI Sec. I C, I D) that modify the probe-measurement CT's to achieve a much broader variety of steady states of the bath.
Remarkably, for the CT M m,0 acting on a spin-chain bath with N = 4 and uniform nearest-neighbor coupling J, we get a completely pure steady state of the bath that is useful for quantum sensing: In contrast, for the CT M 0,n we obtain the mixed state as in Eq. (S21). The purity of this bath for the advantageous CT M m,0 with g 1 τ = π is shown in Supplementary Fig.6. The corresponding bath density matrix is shown in Supplementary Fig.7.  To obtain full purity of the bath, one can switch on resonant polarization transfer between the probe and bath spins according to the Hartmann-Hahn procedure [3][4][5] and then perform a selective measurement on the probe that can polarize the bath completely. If the bath is in the |N/2⟩ state, which is an eigenstate of the resonant S-B (flip-flop) interaction, the probe state remains undisturbed and hence a selective measurement will lead to a deterministic measurement result collapsing the bath to the fully polarized |N/2⟩ = |0, 0 · · · 000⟩ state.